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Gompertz regression parameterized as accelerated failure time model - - PowerPoint PPT Presentation

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Gompertz regression parameterized as accelerated failure time model Filip Andersson and Nicola Orsini Biostatistics Team Departmentof Public Health Sciences Karolinska Institutet 2017 Nordic and Baltic Stata meeting Content Introduction

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Gompertz regression parameterized as accelerated failure time model

Filip Andersson and Nicola Orsini Biostatistics Team Departmentof Public Health Sciences Karolinska Institutet 2017 Nordic and Baltic Stata meeting

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Content

§ Introduction § Proportional hazard model § Accelerated failure time model § The Gompertz distribution § Structural equation models and mediation § Mediation in survival models § Estimating confidence intervals § What I am working on

2017-08-31 Filip Andersson 2

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Content

§ Example

à Data à Pre-estimation à Gompertz proportional hazard à Cox regression à Gompertz vs. Kaplan-Maier à Gompertz ATF model à Post-estimation à Conclusion

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Introduction

§ Why use parametric surival models?

à Can handle right-, left- or interval-censored data à Cox regression can’t handle left- or interval-censored data à Produce better estimation if you have a theoretical expectation of the baseline hazard à Can estimate expected life, not only hazard ratios (AFT-models) à Can include random effects – frailty models (not discussed here)

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Introduction

§ A model that is lacking an easy way to estimate in Stata

à Gompertz regression parameterized as accelerated failure time model à Exist in R

§ eha-package, with command: aftreg

§ Why use Stata?

à Easy handling survival data

§ Data management § Setup

à Good graphical possibility

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Proportional hazard model

§ Easy to compare with Cox regression

à Hazard ratios à Plots

§ Cummulative hazard function § Survival function

à Commonly used

§ Hazard function general form

à ℎ 𝑢 𝑦 = ℎ%(𝑢)𝑓)*

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Accelerated failure time model

§ Can be seen as a linear model (simplest form):

à log 𝑢 = 𝑏 + 𝑐𝑦 + 𝜁 à Usefulin mediation

§ Estimation on life scale

à Estimation of expected baseline life

§ Area under the survival curve when all covariates are zero

à Compare expected life between two groups

§ Logarithmic change in expected life compared to the baseline life expectancy § Expected life = Baseline life expectancy ∗ exp (effect)

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Accelerated failure time model

§ Definiton of accelerated failure time model

à For a group (X1,X2…Xp) , the model is written mathematically as 𝑇 𝑢 𝑦 = 𝑇%

C D()) , where S0(t) is the baseline survival function and

𝜃(𝑦) is an acceleration factor that is a ratio of survival times corresponding to any fixed value of S(t). The acceleration factor is given according to the formula 𝜃 𝑦 = 𝑓(FG)GH⋯HFJ)J). (Qi, J (2009))

§ Hazard function

à ℎ 𝑢 𝑦 =

K D()) ℎ% C D())

§ Log-linear from

à log 𝑢 = 𝑏 + 𝑐𝑦 + 𝜏𝜁 à Where t and ε following corresponing distributions

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The Gompertz distribution

§ When is it useful?

à Adult and old age mortality for humans

§ Demographic models § Including models with treatment effects, such as cancer patiens § Can be problem with very old individuals

§ Normal paramertization

à ℎ 𝑢 = 𝜇𝑓NC à 𝜇 > 0, 𝛿 ≥ 0, 𝑢 > 0

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The Gompertz distribution

§ Suggested new parametrization by Broström, G & Edvinsson, S (2013)

à 𝜇 → U

N , 𝛿 → K N

à ℎ 𝑢 = U

N𝑓

V W

X

à 𝜇 > 0, 𝛿 > 0, 𝑢 > 0

§ Proof of new parametrization

à Hazard for AFT-model à ℎ 𝑢 𝑦 =

K D()) ℎ% C D())

à Here, new gamma can be seen as an accelerated factor

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The Gompertz distribution

§ Linear model: log 𝑢 = 𝑏 + 𝑐𝑦 + 𝜁

§ Here, ε is following a log-Gompertz or inverse Weibull distribution § Compare to the Weibull model, where ε follows a Gumbel distribution

§ Likelihood function

à Survivalfunction: 𝑇 𝑢 = 𝑓𝑦𝑞 −𝜇 𝑓

V W

X − 1

à Density function: 𝐺 𝑢 = ℎ 𝑢 𝑇 𝑢 à Hazard function: ℎ 𝑢 = U

N𝑓

V W

X

à 𝑀 𝛽, 𝜈, 𝜏 = ∏ ℎa 𝑢a 𝑇a(𝑢a) bc 𝑇a(𝑢a) Kdbc

e afK

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Structural equation models and mediation

§ Simple way to estimate linear models within a pathway framework § Estimate all equations and combine for the direct and indirect effects § Supported by most statistical programs

à In Stata the gsem-command combined with simulation is preferable

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Mediation in survival models

§ What do we need to do?

  • 1. Estimate a parametric survivalmodel
  • 2. Estimate the exposure on the mediator

§ First two steps directly from the gsem output

  • 3. Estimate the indirect, direct and total effect
  • 4. Estimate confidence intervals and significance

§ Step three and four can be done with either simulation or delta method § These models are simple for continous mediators, but can be tricky with binary or categorical mediators

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Estimating confidence intervals

§ Simulation

à Boostraping

§ Seems to be the more popular simulation method § Calculate point estimates for the indirect and direct effects § Simulate these point estimates

à Monte carlo simulation

§ More flexible to handle problematic correlations § Not as straight forward

§ Delta method

§ Easiest method and probably most popular § Need a stronger assumption of normality

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What I am working on

§ A Stata command, staftgomp, to estimate the Gompertz regression parameterized as accelerated failure time model similar to what streg does § A post-estimation command that would make it simple to estimate direct, indirect and total effect, with confidence intervals, for survival models

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Example

§ Scanian Economic Demographic Database (Bengtsson, T., Dribe, M. and Svensson, P. (2012)) § Longitudinal historical database

à Data from 17thcentury and onwards à Here, data from individuals born between 1815-1860 are used à Comes from five rural parishes in western Scania à Consist of important life course events as birth and death, but also births of children, marriage or socioeconomic status are recorded

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Data

§ Variables used:

à ”Treatmentvariable”:

§ Approximation of bad early life conditions § Infant mortality rate at the year of birth § High imr vs. low imr (binary) § Years of high diseaseload such as measles, smallpox and whooping cough (Quaranta, L. (2013))

à Parentalsocioeconomic status

§ Socioceconomic status at birth (binary) § Confounder

à Outcome

§ The individuals are followed until death or out-migration.

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Pre-estimation

§ Compare hazard estimations of Gompertz proportional hazard model and Cox regression § Plot survival curve and compare with Kaplan-Maier § If not acceptable test with different survival distribution until the parametric model is acceptable

à Here, we choose Gompertz as it fits good and are supported theoretically for adult mortality

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Gompertz proportional hazard

. streg imr_high ses, dist(gompertz) Gompertz regression -- log relative-hazard form

  • No. of subjects = 3,756 Number of obs = 3,756
  • No. of failures = 880

Time at risk = 19824107 LR chi2(2) = 26.53 Log likelihood = -1773.9194 Prob > chi2 = 0.0000

  • _t | Haz. Ratio
  • Std. Err. z P>|z| [95% Conf. Interval]
  • ------------+----------------------------------------------------------------

imr_high | 1.259023 .0951873 3.05 0.002 1.085624 1.460119 ses | 1.362878 .1010669 4.17 0.000 1.178513 1.576084 _cons | 9.57e-06 8.25e-07 -134.05 0.000 8.08e-06 .0000113

  • ------------+----------------------------------------------------------------

/gamma | .0002332 8.35e-06 27.92 0.000 .0002168 .0002496

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Filip Andersson 19

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Cox regression

. stcox imr_high ses Cox regression -- Breslow method for ties

  • No. of subjects = 3,756 Number of obs = 3,756
  • No. of failures = 880

Time at risk = 19824107 LR chi2(2) = 28.17 Log likelihood = -5889.8259 Prob > chi2 = 0.0000

  • _t | Haz. Ratio
  • Std. Err. z P>|z| [95% Conf. Interval]
  • ------------+----------------------------------------------------------------

imr_high | 1.261686 .0955679 3.07 0.002 1.087617 1.463614 ses | 1.381581 .102833 4.34 0.000 1.194043 1.598573

  • 2017-08-31

Filip Andersson 20

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Gompertz vs. Kaplan-Maier

2017-08-31 Filip Andersson 21

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Gompertz AFT model

. staftgomp imr_high ses Gompertz AFT regression No. of obs = 3756 Log likelihood = -9325.8767 LR chi2(2) = 14.34 Baseline life expectancy = 11669.94 Prob > chi2 = 0.0008

  • _t | Coef. Std. Err. z P>|z| [95% Conf. Interval]
  • ------------+----------------------------------------------------------------

xb | imr_high | -.0496389 .0261027 -1.90 0.057 -.1007992 .0015214 ses | -.0873728 .0273233 -3.20 0.001 -.1409255 -.0338202

  • ------------+----------------------------------------------------------------

bp | lambda | 8.434053 .0451675 186.73 0.000 8.345526 8.522579 gamma | -2.931995 .102271 -28.67 0.000 -3.132442 -2.731547

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Filip Andersson 22

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Post-estimation

. lincom imr_high, eform ( 1) [xb]imr_high = 0

  • _t | exp(b) Std. Err. z P>|z| [95% Conf. Interval]
  • ------------+----------------------------------------------------------------

(1) | .951573 .0248386 -1.90 0.057 .9041146 1.001523

  • . nlcom exp([xb]imr_high)*11699

_nl_1: exp([xb]imr_high)*11699

  • _t | Coef. Std. Err. z P>|z| [95% Conf. Interval]
  • ------------+----------------------------------------------------------------

_nl_1 | 11132.45 290.5866 38.31 0.000 10562.91 11701.99

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Filip Andersson 23

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Post-estimation

§ Baseline life expectancy

à KKghh

igj days = 32,1 years

§ Estimating for individuals after 16000 days

à KKghhHKg%%%

igj

days = 75,9 years of age

§ Effect of high imr during birth

à KKKiqHKg%%%

igj

days = 74,3 years of age

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Conclusion

§ Conclusion

à Even if you survive over the age of 40 you still have a mean shorter life expectancyof 1,6 years if you were born in a year with high imr à Latent effect à Support for the fetal origins hypothesis à Is the estimate reasonable?

§ If needed

à Mediation analysisand calculation ofdirect, indirectand total effect

  • f treatment

à Here, total effect = direct effect

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